PROJECTIVE REPRESENTATIONS OF A QUIVER WITH THREE VERTICES AND TWO EDGES AS R[x]-MODULES

Abstract

In this paper we show that the projective properties of representations of a quiver Q = • → • → • as left R[x]-modules. We show that if P is a projective left R-module then 0 ->0 -> P [x] is a projective representation of a quiver Q as R[x]-modules, but P [x]->0 -> 0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸= 0. And we show a representation 0->P [x]-{id}->P [x] of a quiver Q is a projective representation, if P is a projective left R-module, but P [x]-{id->P [x]-> 0 is not a projective representation of a quiver Q as R[x]-modules, if P ̸=0. Then we show a representation P[x]-{id}->P[x] -{id}-> P[x] of a quiver Q is a projective representation, if P is a projective left R-module.